## anonymous one year ago I need help understanding integration by substitution: 63/(9x+2)^8

1. phi

if you let u = 9x+2 what is du ?

2. xapproachesinfinity

This is the as saying f(x) =9x+2 What is f'(x) Just a difference of differentials but you don't need to worry about thst

3. anonymous

du=9 right?

4. xapproachesinfinity

Yes just one thing you missed Du=9dx

5. phi

no du/dx =9 but if you "take the derivative" u= 9x you differentiate the variables, in this case the u and the x like this du = 9 dx

6. phi

and once you have du = 9 dx then you also have $dx = \frac{1}{9} du$

7. phi

now do your variable substitution replace 9x+2 with u replace dx with 1/9 du

8. phi

and if we have limits, we would also change the limits.

9. phi

and because it is easy to integrate a power, I would use the -8 exponent and get rid of the fraction

10. anonymous

So would it be: $63u ^{-8}*1/9 du$ ??

11. phi

yes and the problem is $7 \int u^{-8} du$ which you can do , right?

12. xapproachesinfinity

Yes

13. phi

once you integrate, replace u with 9x+2 to put the answer in terms of x

14. SolomonZelman

$$\large\color{slate}{\displaystyle\int\limits_{~}^{~}\frac{63}{(9x+2)^8}~dx}$$ $$\large\color{slate}{\displaystyle u=9x+2}$$ $$\large\color{slate}{\displaystyle du=(9x+2)'~\cdot dx~~~\rightarrow~~~du=9~dx}$$ (you already have a 9 and dx to replace that by du, just need to re-write it. $$\large\color{slate}{\displaystyle\int\limits_{~}^{~}\frac{7\cdot 9}{(9x+2)^8}~dx}$$ $$\large\color{slate}{\displaystyle\int\limits_{~}^{~}\frac{7}{(9x+2)^8}~(9\cdot dx)}$$ substitution:: $$\large\color{slate}{\displaystyle\int\limits_{~}^{~}\frac{7}{(u)^8}~(du)}$$ and then all you have left:: $$\large\color{slate}{\displaystyle\int\limits_{~}^{~}7~u^{-8}~du}$$

15. SolomonZelman

(APPLY THE POWER RULE, AND don't forget to substitute the x back for u.)

16. SolomonZelman

igtg....

17. anonymous

Ok, I got it! Thank you all!